The primary objective of the presented study is to investigate the pairwise interaction dynamics between invasive and invaded species, considering a model characterized by a non-regular, non-Lipschitz type reaction, as well as non-homogeneous diffusion. To achieve this, we begin with the foundational model proposed by Bramson in 1988 and tailor it to account for density-dependent diffusion and the non-Lipschitz type reaction, rendering it more applicable to our specific ecological scenario. Subsequently, our newly developed model is subjected to different analyses to ascertain the existence and uniqueness of positive weak solutions. It is noteworthy that density-dependent diffusive operators exhibit a property known as “finite propagation”, which manifests as the existence of a propagating front in the ecological system. Furthermore, we delve into the problem domain by employing the concept of traveling waves to identify specific solutions. A key outcome of our investigation is as follows: When both species propagate at significantly different speeds in the context of traveling waves, the interaction between them is deemed unstable, resulting in oscillations in the concentration of the invaded species. Conversely, when both species propagate within a similar range of speeds, the dynamics of the system are predominantly governed by the invasive species.AMS Subject Classification: 35K55, 35K57, 35K59, 35K65

中文翻译：

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结合具有密度依赖性扩散和非 Lipschitz 反应的 Bramson 模型的侵入-侵入相互作用

本研究的主要目的是研究入侵物种和被入侵物种之间的成对相互作用动力学，考虑以非规则、非利普希茨型反应以及非均匀扩散为特征的模型。为了实现这一目标，我们从 Bramson 于 1988 年提出的基础模型开始，对其进行定制以考虑密度依赖扩散和非 Lipschitz 型反应，使其更适用于我们特定的生态场景。随后，我们新开发的模型进行了不同的分析，以确定正弱解的存在性和唯一性。值得注意的是，密度相关的扩散算子表现出一种称为“有限传播”的特性，表现为生态系统中存在传播前沿。此外，我们通过采用行波的概念来深入研究问题领域，以确定具体的解决方案。我们研究的一个关键结果如下：当两种物种在行波背景下以显着不同的速度传播时，它们之间的相互作用被认为是不稳定的，导致入侵物种浓度的振荡。相反，当两个物种在相似的速度范围内繁殖时，系统的动态主要由入侵物种控制。AMS 主题分类：35K55、35K57、35K59、35K65